I purchased some lower thirds that are 1920 x 1080 and I need to bring them to 1280 x 720, which is usally a fairly easy adjustment - adjust composition settings down, and then deal with shrinking all the elements and layers within the design - but there are no handles. This is the first time I'm dealing with a purchased template that has a "composition within a composition" and I'm on a click hunt to try to figure out what controls, if any I have over this. Double clicking the "comp within the comp" doesn't show me any of the internal elements or details about components.

It's gotta be the most simple thing in the world - shrink down proportionally a composition within the total composition!

Normally when I hear the term "handles", it's referring to the portion of a video clip that has been trimmed up before the in point, and after the out point.

You just want to scale down a 1080 comp to fit properly in a 720 comp. Well, that's a piece o'cake! Nest the 1080 comp inside the 720 comp. Highlight the nested comp. You can go Layer>Scale>Fit to Comp or you can use the keyboard shortcut -- ctl-alt-f (win) or cmd-opt-f (mac).

I think you're my hero right now, but for future reference . . . I found the pathway solution as follows

Layer>TRANSFORM>Fit to Comp

Layer>SCALE - did not provide any Fit to Comp option so I nearly lost hope.

Now I just have to figure out how to get nested comps into PP, as opposed to AE named projects and I'm good to go. I have hundreds of lower thirds with code numerical names and I'm not sure how this impacts workflow.

You can automatically scale to fit any asset in Premiere Pro including a dynamically linked AE composition.

You can also scale to fit horizontally (much better than scale to fit and safer to) any comp that is nested in another comp. Resizing all of the elements separately is, pardon my colloquialism, "nuts"…